Self-induced conversion in dense neutrino gases: Pendulum in flavor space

Neutrino-neutrino interactions can lead to collective flavor conversion effects in supernovae and in the early universe. We demonstrate that the case of bipolar oscillations, where a dense gas of neutrinos and antineutrinos in equal numbers completely converts from one flavor to another even if the mixing angle is small, is equivalent to a pendulum in flavor space. Bipolar flavor conversion corresponds to the swinging of the pendulum, which begins in an unstable upright position (the initial flavor), and passes through momentarily the vertically downward position (the other flavor) in the course of its motion. The time scale to complete one cycle of oscillation depends logarithmically on the vacuum mixing angle. Likewise, the presence of an ordinary medium can be shown analytically to contribute to a logarithmic increase in the bipolar conversion period. We further find that a more complex (and realistic) system of unequal numbers of neutrinos and antineutrinos is analogous to a spinning top subject to a torque. This analogy easily explains how such a system can oscillate in both the bipolar and the synchronized mode, depending on the neutrino density and the size of the neutrino-antineutrino asymmetry. Our simple model applies strictly only to isotropic neutrino gasses. In more general cases, and especially for neutrinos streaming from a supernova core, different modes couple to each other with unequal strength, an effect that can lead to kinematical decoherence in flavor space rather than collective oscillations. The exact circumstances under which collective oscillations occur in nonisotropic media remain to be understood.